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MicroeconomicsGeneralEquilibrium:pricetakingExcessDemandDecentralis-ationandTradeAmany-personeconomyRobinsonCrusoe'sProblemExcessDemandDecentra

lis-ationandTradeRobinsonCrusoe'sProblemGeneralEquilibriumWhyprice-taking?Amany-personeconomyOverview...Sofarwe'vefocusedoncompetitiveequi

libriumanalysis.Butwhy?Whyconcentrateonequilibrium?Whyassumecompetitivebehaviour?Let'sstartbyhavinganotherlookattheexchangeeconomy.We'llredrawtheE

dgeworthboxSomepuzzlesincompetitiveequilibriumanalysisGeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-person

EconomyWhyprice-taking?ExcessdemandThesolutionconceptPricesandtheCoreThesolutionconceptAnexchangeeconomyPricesandtheCoreWhyprice-taking?DEOax2ax1a

DEx1ax2ax1bObOaDEx2bx1bObWhatwouldIdooutofequilibrium?Andwhatwouldhedo?Let'slookattheinnerworkings...BuildinganEdgeworthboxx1aR1aR2

ax2aOaThesituationfromAlf'spointofviewHowAlfrespondsasp1/p2changes...x1aR1aR2ax2aOaAlf'soffercurveWhataboutBill?x1bR1bR2bx2bObThesituationfromhisp

ointofview...asanAustralianx1bR1bR2bx2bObHowBillrespondsasp1/p2changes...x1bR1bR2bx2bObBill'soffercurveThetwooffercurvesx2bOb

x1ax2aOax1bR1bR1aR2aR2bOffersareonlyconsistentwherecurvesintersectBut,byconstruction...Thisisthecompetitiveequil

ibriumSatisfiesmaterialsbalanceEDequilibriumhereendowmentherePrice-takingutility-maximisingBillendowmenthereequilibriumherePrice-takingutilit

y-maximisingAlf•Butwhycompetitivebehaviour?•WhyshouldAlfandBillbehavethatway?•Wheredothepricescomefrom?Givencompetitivebehaviour,theCEisthe

only"consistent"allocationsubjecttohhSpixiMi=1U(x)maxnhhSpixiMSowheredothepricescomefrom?Weneedamoregeneralsolut

ionconceptThenexaminehowprice-takingequilibriumrelatestothis.Let'sdeveloptheapproachfortheexchangeeconomy,thoughitcouldapplytomoreinterest

ingeconomies.AfreshapproachGeneralequilibriumRobinsonCrusoe’sproblemDecentrali-sationandTradeAmany-personEconomyWhy

price-taking?ExcessdemandThesolutionconceptPricesandtheCoreAnexchangeeconomyPricesandtheCoreAnexchangeeconomyWhyprice-taking?Viewthepopulation

...Q2Q1...asnseparateindividualsQ0Acoalition...Thecoalitionconceptgivesusanimportantclue....isformedbyanysubgroupwe'vegotourendowmentswithusO

nedayyoutakeyourbundlealongtothe"swapshop"......andsomebossypersonistherewhoproposes(insistson?)aparticularfea

sibleallocation.Youandsomeothersdon'tlikethexbundleyouwouldgetunderthisallocation......andyoufindthatbyusingjusttheresourcesyouallowny

oucouldallgetautilitylevelatleastashighasthatofferedtoyouundertheproposedallocation.Youguysthereforerefusetoacce

ptit.Theideaofblocking:alittlestoryAnallocationisblocked...Q2^xb^xc^xd^xa^xe^xf^xg^xl^xk...whichcandobett

erforitselfifwecanfindacoalition...AproposedallocationforthewholeeconomyhhU(x)^hhU(x)...andthatarefeasibleQ{1,2,...,nh}^h[x]isblockedbyQ"hQ,">"f

orsomehhQhSxhSRhQAcoalitionBundlesthatarepreferredbymembersofthecoalition...^h[x]AformalapproachTheCoreisthesetofunblocked,feasibleallocat

ionsUsetheideaofblockingtointroduceabasicsolutionconcept{Alf&Bill}{Alf}{Bill}Ina2-personworldthereareveryfewcoalitionspossible

UsetheEdgeworthbox...x2bObx1ax2aOax1b...togetthecontractcurve{Alf}blockstheseallocations...D{Bill}blockstheseallocations

...{Alf,Bill}(together)blockthese...EndowmentNo-onecanblockanallocationhereThe2-personcoreGeneralequilibriumRobinsonCrusoe’sproblem

Decentrali-sationandTradeAmany-personEconomyWhyprice-taking?ExcessdemandThesolutionconceptPricesandtheCoreAnexchangeeco

nomyThesolutionconceptAnexchangeeconomyWhyprice-taking?x2bObx1ax2aOax1bDEThecompetitiveequilibriumagainx2bObx1ax2aOax1bDEAnycompetitiveequilibr

iummustlieinthecoreEveryCEallocationmustbelongtothecoreOfcourseitispossiblethatnoCEexists(seenextlecture)Butwhatofothercoreallocationswhichareno

tCE?Let'stakeaclose-uplookAsimpleresultDBAlfBillAlfBillATheextremesofthe2-personcoreCoreCEIntroducingAlfandBill'slong

-losttwinbrothers...Rememberwearedealingwitha2-personmodelButwilltherealwaysbenon-CEpointsinthecore?Abasicre

lation...AlfandhistwinbrotherArthurhavethesamepreferencesandendowmentsLikewisethetwinsBillandBenNowofcoursetherearemorepossibilitieso

fformingcoalitionsSolet'sclonetheeconomy{Arthur&Ben}{Arthur}{Ben}{Alf&Arthur}{etc,etc}{Bill&Ben}{Alf&Bill}{Alf

}{Bill}{Alf,Arthur&Bill}...plussomenewonesAlltheoldcoalitionsarepossible...Willthesestillbecoreallocationsinthe4-perso

neconomy?BAAlf'n'ArthurBill'n'BenAlf'n'ArthurBill'n'BenDAlf,Arthur,BenBillG...leavingtheBentwinoutsidethecoalitionThisalloca

tionisnotasolution...ButitshowsthatthecorehasbecomesmallerNo.{Alf-Arthur-Bill}canblockA...Alf½[xa+Ra](G)Arthur½[xa+Ra](G)Billxb(A)Be

nRb(D)2Ra+RbThecoalitionFeasibleHowtheblockingcoalitionworksABDCanthisallocationbeblocked?AnA-personwouldbebetteroffsomewherehereBingo!Whatif

therewere500ofeachtype...?500oftheAtribe450oftheBtribe500oftheAtribe400oftheBtribe500oftheAtribe360oftheBtribe500oftheAtribe310oftheBtribe500of

theAtribe250oftheBtribeIfNisbiggertheremaybemoreblockingcoalitionsABDETheonlyallocationleftinthe2N-personcore(eventually)...Asyouclonethee

conomythecorebecomessmaller.Inthelimititcontainsonlycompetitiveequilibriumallocations.Butnoticethatthisrequiresbalanc

edreplicationoftheAlfandBilltribesAnditassumesthatallthesevariouscoalitionsaresomehowrelevantApowerfulresult:theshrinkingcoreEndowme

ntsPreferencesAllocationsCoalitionsBlockingAnswertoquestion:whyprice-taking?Inalargeeconomywithsuitablysmallagents.......it'stheonlythingtodo.Afin

alreminder:Basicconceptsofthecore•Aggregatingfirms•Aggregatingconsumers•WARP•Propertiesofdemandsystems•TheSeparationTheorem•Firm#4slides18

-26•Consumer#4slides39-44•Consumer#1slides23.24•Consumer#3slides5,6•GenEqm#2slides14-20,24,25.